Electromagnetic waves have long been used to measure the saturation and salinity of water in porous rocks. The electromagnetic measurements use propagating radio waves and determine the phase and amplitude of the received signal for known transmitter-receiver geometries and operating frequencies. These phase and amplitude measurements are independent; they are converted to apparent dielectric permittivity and electric conductivity for some tools, such as the Electromagnetic Propagation Tool (EPT) or the Deep Propagation Tool (DPT), both of Schlumberger.
Electromagnetic-propagation measurements were invented and first introduced in the formation-evaluation industry in the 1970s. Several generations and implementations of these tools used similar algorithms to convert the measured phases and amplitudes to apparent resistivities or to invert them to apparent dielectric permittivity and electric conductivity. Examples of such techniques can be found in U.S. Pat. No. 3,944,910, U.S. Pat. No. 4,704,581 and U.S. Pat. No. 4,899,112.
Many electromagnetic measurement devices use a differential receiver configuration. Two receivers are placed at known (rnear and rfar) distances from the transmitter and with known relative orientation to the transmitter dipole moment (typically aligned). The differential signal V simply is the ratio of the two raw signals. The phase difference between the two receivers is called the phase shiftPS(=φfar−φnear),and the difference of the amplitude logarithms (or logarithmic amplitudes) is called attenuationAT(=αnear−αfar).
The purpose of these electromagnetic measurements is to determine the electric conductivity of the surrounding medium from the observed phase and amplitude, or phase shift and attenuation. The dielectric measurement determines the two unknowns: electric conductivity and dielectric permittivity simultaneously from the two measurements: phase shift and attenuation.
Commonly, the measurements are converted to an apparent electric conductivity (or resistivity R=1/σ) using interpolation in pre-computed look-up tables. The simultaneous conversion of phase shift and attenuation into apparent dielectric permittivity and electric conductivity requires a two-dimensional look-up table. Such tables have been published and are widely accepted in the industry.
As an alternative approach, the measured phase shift and attenuation data can be inverted directly for permittivity ∈ and conductivity σ as is discussed in ANDERSON, Barbara, et al. OBSERVATIONS OF LARGE DIELECTRIC EFFECTS ON LWD PROPAGATION-RESISTIVITY LOGS. SPWLA 48th Annual Logging Symposium. 3-6 Jun. 2007. The iterative inversion algorithm minimizes the sum of the squared differences between the measured quantities and computed signals that are functions of the relative dielectric permittivity and the electric conductivity
  L  =            1      2        ⁢          (                                                                                    (                                                            PS                      meas                                        -                                                                  PS                        simul                                                  (                          n                          )                                                                    ⁡                                              (                                                                              ɛ                            r                                                          (                              n                              )                                                                                ,                                                      σ                                                          (                              n                              )                                                                                                      )                                                                              )                                2                            +                                                                                          (                                                      AT                    meas                                    -                                                            AT                      simul                                              (                        n                        )                                                              ⁢                                                                                  ⁢                                          (                                                                        ɛ                          r                                                      (                            n                            )                                                                          ,                                                  σ                                                      (                            n                            )                                                                                              )                                                                      )                            2                                          )      
The values for permittivity and conductivity are iteratively updated (by the index n) to reconcile the simulated with the measured data and thus arrive at the best estimate for the relative dielectric permittivity and the electric conductivity. The n-th iteration for the permittivity and conductivity are calculated from the (n−1)-th iteration by a linear correction
      (                                        ɛ            r                          (              n              )                                                                        σ                          (              n              )                                            )    =            (                                                  ɛ              r                              (                                  n                  -                  1                                )                                                                                        σ                              (                                  n                  -                  1                                )                                                        )        +                  (                                                                              ∂                                      ɛ                    r                                                                    ∂                  PS                                                                                                      ∂                                      ɛ                    r                                                                    ∂                  AT                                                                                                                          ∂                  σ                                                  ∂                  PS                                                                                                      ∂                  σ                                                  ∂                  AT                                                                    )            ⁢              (                                                                              PS                  meas                                -                                  PS                  simul                                      (                                          n                      -                      1                                        )                                                                                                                                            AT                  meas                                -                                  AT                  simul                                      (                                          n                      -                      1                                        )                                                                                      )            
Previous implementations of this standard inversion algorithm always computed the four partial derivatives independently. These computations are the most time-consuming step in the iteration.
This invention aims to provide a computationally less intensive approach that uses the phase shift and attenuation measurements to derive a complex-valued quantity, and uses this complex-valued quantity for computations of the parameters of interest.